Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 150539, 9 pages
doi:10.1155/2010/150539
Research Article
Strong Convergence Theorems for Strict
Pseudocontractions in Uniformly Convex
Banach Spaces
Liang-Gen Hu,1 Wei-Wei Lin,2 and Jin-Ping Wang1
1
2
Department of Mathematics, Ningbo University, Zhejiang 315211, China
School of Computer Science and Engineering, South China University of Technology,
Guangzhou 510640, China
Correspondence should be addressed to Liang-Gen Hu, hulianggen@yahoo.cn
Received 20 April 2010; Accepted 26 August 2010
Academic Editor: W. Takahashi
Copyright q 2010 Liang-Gen Hu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The viscosity approximation methods are employed to establish strong convergence theorems of
the modified Mann iteration scheme to λ-strict pseudocontractions in p-uniformly convex Banach
spaces with a uniformly Gâteaux differentiable norm. The main result improves and extends many
nice results existing in the current literature.
1. Introduction
Let E be a real Banach space, and let C be a nonempty closed convex subset E. We denote by
∗
J the normalized duality map from E to 2E defined by
Jx x∗ ∈ E∗ : x, x∗ x2 x∗ 2 , ∀x ∈ E .
1.1
A mapping T : C → C is said to be a λ-strictly pseudocontractive mapping see, e.g.,
1 if there exists a constant 0 ≤ λ < 1 such that
T x − T y2 ≤ x − y2 λI − T x − I − T y2 ,
1.2
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Fixed Point Theory and Applications
for all x, y ∈ C. We note that the class of λ-strict pseudocontractions strictly includes the class
of nonexpansive mappings which are mapping T on C such that
T x − T y ≤ x − y,
1.3
for all x, y ∈ C. Obviously, T is nonexpansive if and only if T is a 0-strict pseudocontraction.
A mapping T : C → C is said to be a λ-strictly pseudocontractive mapping with respect to
p if, for all x, y ∈ C, there exists a constant 0 ≤ λ < 1 such that
T x − T yp ≤ x − yp λI − T x − I − T yp .
1.4
A mapping f : C → C is called k-contraction if there exists a constant k ∈ 0, 1 such that
fx − f y ≤ kx − y,
∀x, y ∈ C.
1.5
We denote by FixT the set of fixed point of T , that is, FixT {x ∈ C : T x x}.
Recall the definition of Mann’s iteration; let C be a nonempty convex subset E, and let T
be a self-mapping of C. For any x1 ∈ C, the sequence {xn } is defined by
xn1 1 − αn xn αn T xn ,
n ≥ 1,
1.6
where {αn } is a real sequence in 0, 1.
In the last ten years or so, there have been many nice papers in the literature
dealing with the iteration approximating fixed points of Lipschitz strongly pseudocontractive
mappings by utilizing the Mann iteration process. Results which had been known only for
Hilbert spaces and Lipschitz mappings have been extended to more general Banach spaces
and more general class of mappings; see, for example, 1–6 and the references therein for
more information about this problem.
In 2007, Marino and Xu 2 showed that the Mann iterative sequence converges
weakly to a fixed point of λ-strict pseudocontractions in Hilbert spaces. Meanwhile, they
have proposed an open question; that is, is the result of 2, Theorem 3.1 true in uniformly convex
Banach spaces with Fréchet differentiable norm? In other words, can Reich’s theorem 7, Theorem
2, with respect to nonexpansive mappings, be extended to λ-strict pseudocontractions in
uniformly convex Banach spaces?
In 2008, using the Mann iteration and the modified Ishikawa iteration, Zhou 3
obtained some weak and strong convergence theorems for λ-strict pseudocontractions in
Hilbert spaces which extend the corresponding results in 2.
Recently, Hu and Wang 4 obtained that the Mann iterative sequence converges
weakly to a fixed point of λ-strict pseudocontractions with respect to p in p-uniformly convex
Banach spaces.
In this paper, we first introduce the modified Mann iterative sequence. Let C be a
nonempty closed convex subset of E, and let f : C → C be a k-contraction. For any x1 ∈ C, the
sequence {xn } is defined by
xn1 αn xn 1 − αn Tn βn fxn 1 − βn xn ,
n ≥ 1,
1.7
Fixed Point Theory and Applications
3
where Tn x : 1 − μn x μn T x, for all x ∈ C, {αn }, {βn }, and {μn }in 0, 1. The iterative sequence
1.7 is a natural generalization of the Mann iterative sequences 1.6. If we take βn ≡ 0, for
all n ≥ 1, in 1.7, then 1.7 is reduced to the Mann iteration.
The purpose in this paper is to show strong convergence theorems of the modified
Mann iteration scheme for λ-strict pseudocontractions with respect to p in p-uniformly
convex Banach spaces with uniformly Gateaux differentiable norm by using viscosity
approximation methods. Our theorems improve and extend the comparable results in
the following four aspects: 1 in contrast to weak convergence results in 2–4, strong
convergence theorems of the modified Mann iterative sequence are obtained in p-uniformly
convex Banach spaces; 2 in contrast to the results in 7, 8, these results with respect to
nonexpansive mappings are extended to λ-strict pseudocontractions with respect to p; 3
∞
the restrictions ∞
n1 |αn1 − αn | < ∞ and
n1 |βn1 − βn | < ∞ in 8, Theorem 3.1 are removed;
4 our results partially answer the open question.
2. Preliminaries
The modulus of convexity of E is the function δE : 0, 2 → 0, 1 defined by
x y
: x 1, y 1, x − y ≥ ,
δE inf 1 − 2 0 ≤ ≤ 2.
2.1
E is uniformly convex if and only if, for all 0 < ≤ 2 such that δE > 0. E is said to be
p-uniformly convex if there exists a constant a > 0 such that δE ≥ ap . Hilbert spaces, Lp or
p
lp spaces 1 < p < ∞ and Sobolev spaces Wm 1 < p < ∞ are p-uniformly convex. Hilbert
spaces are 2-uniformly convex, while
p
L , l
p
p
, Wm
are
⎧
⎨2-uniformly convex
if 1 < p ≤ 2,
⎩p-uniformly convex
if p ≥ 2.
2.2
A Banach space E is said to have Gateaux differentiable norm if the limit
x ty − x
lim
t→0
t
2.3
exists for each x, y ∈ U, where U {x ∈ E : x 1}. The norm of E is a uniformly G
ateaux
differentiable if for each y ∈ U, the limit is attained uniformly for x ∈ U. It is well known that
if E is a uniformly Gateaux differentiable norm, then the duality mapping J is single valued
and norm-to-weak∗ uniformly continuous on each bounded subset of E.
Lemma 2.1 see 4. Let E be a real p-uniformly convex Banach space, and let C be a nonempty
closed convex subset of E. Let T : C → C be a λ-strict pseudocontraction with respect to p, and let
{ξn } be a real sequence in 0, 1. If Tn : C → C is defined by Tn x : 1 − ξn x ξn T x, for all x ∈ C,
then for all x, y ∈ C, the inequality holds
Tn x − Tn yp ≤ x − yp − wp ξn cp − ξn k I − T x − I − T yp ,
2.4
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Fixed Point Theory and Applications
where cp is a constant in [9, Theorem 1]. In addition, if 0 ≤ λ < min{1, 2−p−2 cp }, ξ 1−λ·2p−2 /cp ,
and ξn ∈ 0, ξ, then Tn x − Tn y ≤ x − y, for all x, y ∈ C.
Lemma 2.2 see 10. Let {xn } and {yn } be bounded sequences in a Banach space E such that
xn1 αn xn 1 − αn yn ,
n ≥ 0,
2.5
where {αn } is a sequence in 0, 1 such that 0 < lim infn → ∞ αn ≤ lim supn → ∞ αn < 1. Assuming
lim sup yn1 − yn − xn1 − xn ≤ 0,
2.6
n→∞
then limn → ∞ xn − yn 0.
Lemma 2.3. Let E be a real Banach space. Then, for all x, y ∈ E and jxy ∈ Jxy, the following
inequality holds:
x y2 ≤ x2 2 y, j x y .
2.7
Lemma 2.4 see 11. Let {an } be a sequence of nonnegative real number such that
an1 ≤ 1 − δn an δn ηn ,
∀n ≥ 0,
2.8
where {δn } is a sequence in 0, 1 and {ηn } is a sequence in R satisfying the following conditions:
∞
i ∞
n1 δn ∞; ii lim supn → ∞ ηn ≤ 0 or
n1 δn |ηn | < ∞. Then, limn → ∞ an 0.
3. Main Results
Theorem 3.1. Let E be a real p-uniformly convex Banach space with a uniformly G
ateaux
differentiable norm, and let C be a nonempty closed convex subset of E which has the fixed point
property for nonexpansive mappings. Let T : C → C be a λ-strict pseudocontraction with respect to
∅. Let f : C → C be a k-contraction with k ∈ 0, 1.
p, λ ∈ 0, min{1, 2−p−2 cp } and FixT /
Assume that real sequences {αn }, {βn }, and {ξn } in 0, 1 satisfy the following conditions:
i 0 < lim infn → ∞ αn ≤ lim supn → ∞ αn < 1,
ii limn → ∞ βn 0 and ∞
n1 βn ∞,
iii 0 < infn ξn ≤ ξ and limn → ∞ |ξn1 − ξn | 0, where ξ 1 − λ · 2p−2 /cp .
For any x1 ∈ C, the sequence {xn } is generated by
xn1 αn xn 1 − αn Tn βn fxn 1 − βn xn ,
n ≥ 1,
3.1
where Tn x : 1 − ξn x ξn T x, for all x ∈ C. Then, the sequence {xn } converges strongly to a fixed
point of T .
Fixed Point Theory and Applications
5
Proof. Equation 3.1 can be expressed as follows:
xn1 αn xn 1 − αn Tn yn ,
3.2
where
yn βn fxn 1 − βn xn ,
∀n ≥ 1.
3.3
Taking p ∈ FixT , we obtain from Lemma 2.1
xn1 − p ≤ αn xn − p 1 − αn Tn yn − p
≤ αn xn − p 1 − αn βn fxn − p 1 − βn xn − p
≤ αn xn − p 1 − αn βn kxn − p βn f p − p 1 − βn xn − p
1 f p − p 1 − 1 − αn βn 1 − k xn − p 1 − αn βn 1 − k
1−k
1 ≤ max x1 − p,
f p − p .
1−k
3.4
Therefore, the sequence {xn } is bounded, and so are the sequences {fxn }, {Tn yn }, and {yn }.
Since Tn yn 1 − ξn yn ξn T yn and the condition iii, we know that {T yn } is bounded. We
estimate from 3.3 that
yn1 − yn ≤ βn1 fxn1 − fxn 1 − βn1 xn1 − xn βn1 − βn fxn − xn ≤ 1 − βn1 1 − k xn1 − xn βn1 − βn fxn − xn .
3.5
Since Tn : 1 − ξn I ξn T , where I is the identity mapping, we have
Tn1 yn1 − Tn yn ≤ 1−ξn1 yn1 ξn1 T yn1 −1−ξn1 yn −ξn1 T yn |ξn1 −ξn |yn −T yn ≤ yn1 − yn |ξn1 − ξn |yn − T yn .
3.6
limn → ∞ βn 0 and limn → ∞ |ξn1 − ξn | 0 imply from 3.5 and 3.6 that
lim sup Tn1 yn1 − Tn yn − xn1 − xn ≤ 0.
n→∞
3.7
Hence, by Lemma 2.2, we obtain
lim Tn yn − xn 0.
n→∞
3.8
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Fixed Point Theory and Applications
From 3.3, we get
lim yn − xn lim βn fxn − xn 0,
n→∞
n→∞
3.9
and so it follows from 3.8 and 3.9 that limn → ∞ yn −Tn yn 0. Since yn −Tn yn ξn yn −T yn and infn ξn > 0, we have
yn − Tn yn lim yn − T yn lim
0.
n→∞
n→∞
ξn
3.10
For any δ ∈ 0, ξ, defining Tδ : 1 − δI δT , we have
lim yn − Tδ yn lim δyn − T yn 0.
n→∞
n→∞
3.11
Since Tδ is a nonexpansive mapping, we have from 12, Theorem 4.1 that the net {xt }
generated by xt tfxt 1 − tTδ xt converges strongly to q ∈ FixTδ FixT , as t → 0.
Clearly,
xt − yn 1 − t Tδ xt − yn t fxt − yn .
3.12
In view of Lemma 2.3, we find
xt − yn 2 ≤ 1 − t2 Tδ xt − yn 2 2t fxt − yn , J xt − yn
2
≤ 1 − 2t t2 xt − yn Tδ yn − yn 2t fxt − xt , J xt − yn
3.13
2
2txt − yn ,
and hence
fxt − xt , J yn − xt
2
1 t2 yn − Tδ yn t
xt − yn ≤
2xt − yn yn − Tδ yn .
2
2t
3.14
Since the sequences {yn }, {xt }, and {Tδ yn } are bounded and limn → ∞ yn − Tδ yn /2t 0, we
obtain
t
lim sup fxt − xt , J yn − xt ≤ M,
2
n→∞
3.15
where M supn≥1,t∈0,1 {xt − yn 2 }. We also know that
f q − q, J yn − q fxt − xt , J yn − xt f q − fxt xt − q, J yn − xt
f q − q, J yn − q − J yn − xt .
3.16
Fixed Point Theory and Applications
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From the facts that xt → q ∈ FixT , as t → 0, {yn } is bounded, and the duality mapping J is
norm-to-weak∗ uniformly continuous on bounded subset of E, it follows that
f q − q, J yn − q − J yn − xt −→ 0,
f q − fxt xt − q, J yn − xt −→ 0,
as t −→ 0,
as t −→ 0.
3.17
Combining 3.15, 3.16, and the two results mentioned above, we get
lim sup f q − q, J yn − q ≤ 0.
n→∞
3.18
From 3.9 and the fact that the duality mapping J is norm-to-weak∗ uniformly continuous
on bounded subset of E, it follows that
lim fxn − f q , J yn − q − J xn − q 0.
3.19
xn1 − q αn xn − q 1 − αn Tn yn − q ,
3.20
n→∞
Writing
and from Lemma 2.3, we find
xn1 − q2 ≤ αn xn − q2 1 − αn βn fxn − q 1 − βn xn − q 2
2
2
2 ≤ αn xn − q 1 − αn 1 − βn xn − q
21 − αn βn fxn − q, J yn − q
2
2
2
2 ≤ αn xn − q 1 − αn 1 − βn xn − q 21 − αn βn kxn − q
21 − αn βn f q − q, J yn − q
21 − αn βn fxn − f q , J yn − q − J xn − q
2
≤ 1 − 21 − αn 1 − kβn xn − q 21 − αn βn
× βn xn −q fxn −f q , J yn −q −J xn −q f q −q, J yn −q
2
1 − 1 − kδn xn − q δn ηn ,
3.21
where
δn 21 − αn βn ,
ηn βn xn − q fxn − f q , J yn − q − J xn − q f q − q, J yn − q .
3.22
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Fixed Point Theory and Applications
∞
∞ and
From 3.18, 3.19, and the conditions i, ii, it follows that
n1 δn
lim supn → ∞ ηn ≤ 0. Consequently, applying Lemma 2.4 to 3.21, we conclude that
limn → ∞ xn − q 0.
Corollary 3.2. Let E, C, T , {αn }, {βn }, and {ξn } be as in Theorem 3.1. For any u, x1 ∈ C, the
sequence {xn } is generated by
xn1 αn xn 1 − αn Tn βn u 1 − βn xn ,
n ≥ 1,
3.23
where Tn x : 1 − ξn x ξn T x, for all x ∈ C. Then the sequence {xn } converges strongly to a fixed
point of T .
Remark 3.3. Theorem 3.1 and Corollary 3.2 improve and extend the corresponding results in
2–4, 7, 8 essentially since the following facts hold.
1 Theorem 3.1 and Corollary 3.2 give strong convergence results in p-uniformly
convex Banach spaces for the modification of Mann iteration scheme in contrast
to the weak convergence result in 2, Theorem 3.1, 3, Theorem 3.1 and Corollary
3.3, and 4, Theorems 3.2 and 3.3.
2 In contrast to the results in 7, Theorem 2, and 8, Theorem 3.1, these results with
respect to nonexpansive mappings are extended to λ-strict pseudocontraction in
p-uniformly convex Banach spaces.
3 In contrast to the results in 8, Theorem 3.1, the restrictions ∞
n1 |αn1 − αn | < ∞
∞
and n1 |βn1 − βn | < ∞ are removed.
Acknowledgments
The authors would like to thank the referees for the helpful suggestions. Liang-Gen Hu was
supported partly by Ningbo Natural Science Foundation 2010A610100, the NNSFC
60872095, the K. C. Wong Magna Fund of Ningbo University and the Scientific
Research Fund of Zhejiang Provincial Education Department Y200906210. Wei-Wei Lin
was supported partly by the Fundamental Research Funds for the Central Universities,
SCUT20092M0103. Jin-Ping Wang were supported partly by the NNSFC60872095 and
Ningbo Natural Science Foundation 2008A610018.
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